Complex Analysis

f(z)=z^2

Complex functions


Let $S$ be a set of complex numbers. A function $f$ defined on $S$ is a rule that assigns to each $z$ in $S$ a complex number $w$. The number $w$ is called the value of $f$ at $z$ and is denoted by $f (z)$; that is, $w = f (z)$. The set $S$ is called the domain of definition of $f$.

If only one value of $w$ corresponds to each value of $z$, we say that $w$ is a single-valued function of $z$ or that $f(z)$ is single-valued. If more than one value of $w$ corresponds to each value of $z$, we say that $w$ is a multiple-valued or many-valued function of $z$.

A multiple-valued function can be considered as a collection of single-valued functions, each member of which is called a branch of the function. In general, we consider one particular member as a principal branch of the multiple-valued function and the value of the function corresponding to this branch as the principal value.

Example 1: The function $w=z^2$ is a single-valued function of $z$. On the other hand, if $w=z^{\frac{1}{2}},$ then to each value of $z$ there are two values of $w$. Hence, the function $$w=z^{\frac{1}{2}}$$ is a multiple-valued (in this case two-valued) function of $z$.

Suppose that $w=u+iv$ is the value of a function $f$ at $z= x+iy$, so that $$u+iv=f(x+iy)$$ Each of the real numbers $u$ and $v$ depends on the real variables $x$ and $y$, and it follows that $f(z)$ can be expressed in terms of a pair of real-valued functions of the real variables $x$ and $y$: \begin{eqnarray}\label{eq1} f(z)= u(x,y)+iv(x,y). \end{eqnarray} If the polar coordinates $r$ and $\theta$, instead of $x$ and $y$ , are used, then $$u+iv=f\left(re^{i\theta}\right)$$ where $w=u+iv$ and $z=re^{i\theta}$. In this case, we write \begin{eqnarray}\label{eq2} f(z)=u\left(r, \theta\right)+iv\left(r, \theta\right). \end{eqnarray}

Example 2: If $f(z)= z^2$ then $$f(x+iy)=(x+iy)^2=x^2-y^2+i(2xy).$$ Hence $$u(x,y)= x^2-y^2\quad \text{and}\quad v(x,y)= 2xy.$$ When we use polar coordinates, we have $$u\left(r, \theta\right)= r^2\cos 2\theta \quad \text{and}\quad v\left(r, \theta\right)= r^2\sin 2\theta.$$

Question: What happens when in either of equations (\ref{eq1}) and (\ref{eq2}) the function $v$ always has a value zero?


Examples of complex functions


Polynomial functions

For $a_n, a_{n-1}, \ldots, a_0$ complex constants we define $$p(z) = a_n z^n + a_{n-1} z^{n-1} +\cdots +a_{1}z + a_0$$ where $a_n\neq 0$ and $n$ is a positive integer called the degree of the polynomial $p(z)$.

Rational functions: Ratios $$\frac{p(z)}{q(z)}$$ where $p(z)$ and $q(z)$ are polynomials and $q(z)\neq 0$.

Exponential function

Exponential function: If $z=x+iy$, the exponential function $e^z$ is defined by writing \begin{eqnarray*} e^z=e^xe^{iy}. \end{eqnarray*} Because \begin{eqnarray*} e^{iy}=\cos y +i\sin y, \end{eqnarray*} then we have \begin{eqnarray*} e^z=e^x\left(\cos y +i\sin y\right). \end{eqnarray*}

Logarithmic function

In a similar fashion, the complex logarithm is a complex extension of the usual real natural (i.e., base $e$) logarithm. In terms of polar coordinates $z = r e^{i\theta}$, the complex logarithm has the form $$\log z = \log\left(r e^{i\theta}\right) = \log r + \log e^{i\theta}  = \log r + i \theta.$$ We will explore in detail this function in the following section.

Trigonometric functions

The sine and cosine of a complex variable $z$ are defined as follows: \begin{eqnarray*} \sin z=\frac{e^{i z}-e^{-iz}}{2i}\quad \text{and}\quad \cos z=\frac{e^{iz} +e^{-i z}}{2}. \end{eqnarray*} The other four trigonometric functions are defined in terms of the sine and cosine functions with the following relations: \begin{align*} \tan z&=\frac{\sin z}{\cos z} & \cot z&=\frac{\cos z}{\sin z} \\ \sec z&=\frac{1}{\cos z} & \csc z&=\frac{1}{\sin z}. \end{align*}

Hyperbolic trigonometric functions

The hyperbolic sine and the hyperbolic cosine of a complex variable are defined as they are with a real variable; that is, \begin{eqnarray*} \sinh z=\frac{e^{z}-e^{-z}}{2}\quad \text{and}\quad \cosh z=\frac{e^{z} +e^{- z}}{2}. \end{eqnarray*} The other four hyperbolic functions are defined in terms of the hyperbolic sine and cosine functions with the relations: \begin{align*} \tanh z&=\frac{\sinh z}{\cosh z} & \coth z&=\frac{\cosh z}{\sinh z} \\ \text{sech } z&=\frac{1}{\cosh z} & \text{csch } z&=\frac{1}{\sinh z}. \end{align*}


Explore the real and imaginary components

Use the following applet to explore the real and imaginary components of some complex functions


Sorry, the applet is not supported for small screens. Rotate your device to landscape. Or resize your window so it's more wide than tall.

Code

Enter the following scripts in GeoGebra to explore it yourself. Open the 3D view. The symbol # indicates comments.

#Define complex function
    
f(z) := z + 1/z

#Define components

Re = Surface(u, v, real( f(u + ί v) ), u, -5, 5, v, -5, 5)

Im = Surface(u, v, imaginary( f(u + ί v) ), u, -5, 5, v, -5, 5)

NEXT: The Logarithmic Function